NeurIPS 2020 Papers

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study s-sparse functions of the form $f(x) = \sum_{j=1}^s a_j e^{i \lambda_j x}$ for coefficients $a_j \in C$ and frequencies $\lambda_j \in R$. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the *statistical dimension* of the approximated kernel matrix. It is the first ``oblivious sketching method'' with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP has impractically slow convergence to high frequency signal components. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

Self-supervised depth estimators have recently shown results comparable to the supervised methods on the challenging single image depth estimation (SIDE) task, by exploiting the geometrical relations between target and reference views in the training data. However, previous methods usually learn forward or backward image synthesis, but not depth estimation, as they cannot effectively neglect occlusions between the target and the reference images. Previous works rely on rigid photometric assumptions or on the SIDE network to infer depth and occlusions, resulting in limited performance. On the other hand, we propose a method to "Forget About the LiDAR" (FAL), with Mirrored Exponential Disparity (MED) probability volumes for the training of monocular depth estimators from stereo images. Our MED representation allows us to obtain geometrically inspired occlusion maps with our novel Mirrored Occlusion Module (MOM), which does not impose a learning burden on our FAL-net. Contrary to the previous methods that learn SIDE from stereo pairs by regressing disparity in the linear space, our FAL-net regresses disparity by binning it into the exponential space, which allows for better detection of distant and nearby objects. We define a two-step training strategy for our FAL-net: It is first trained for view synthesis and then fine-tuned for depth estimation with our MOM. Our FAL-net is remarkably light-weight and outperforms the previous state-of-the-art methods with 8$\times$ fewer parameters and 3$\times$ faster inference speeds on the challenging KITTI dataset. We present extensive experimental results on the KITTI, CityScapes, and Make3D datasets to verify our method's effectiveness. To the authors' best knowledge, the presented method performs the best among all the previous self-supervised methods until now.

We address the problem of credit assignment in reinforcement learning and explore fundamental questions regarding the way in which an agent can best use additional computation to propagate new information, by planning with internal models of the world to improve its predictions. Particularly, we work to understand the gains and peculiarities of planning employed as forethought via forward models or as hindsight operating with backward models. We establish the relative merits, limitations and complementary properties of both planning mechanisms in carefully constructed scenarios. Further, we investigate the best use of models in planning, primarily focusing on the selection of states in which predictions should be (re)-evaluated. Lastly, we discuss the issue of model estimation and highlight a spectrum of methods that stretch from environment dynamics predictors to planner-aware models.

We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ \emph{worst case} regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is \emph{predictable}, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of $O(T^{-1/2})$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making $O(T^{1/2})$ parallel calls to the optimization oracle.

Unsupervised learning from continuous visual streams is a challenging problem that cannot be naturally and efficiently managed in the classic batch-mode setting of computation. The information stream must be carefully processed accordingly to an appropriate spatio-temporal distribution of the visual data, while most approaches of learning commonly assume uniform probability density. In this paper we focus on unsupervised learning for transferring visual information in a truly online setting by using a computational model that is inspired to the principle of least action in physics. The maximization of the mutual information is carried out by a temporal process which yields online estimation of the entropy terms. The model, which is based on second-order differential equations, maximizes the information transfer from the input to a discrete space of symbols related to the visual features of the input, whose computation is supported by hidden neurons. In order to better structure the input probability distribution, we use a human-like focus of attention model that, coherently with the information maximization model, is also based on second-order differential equations. We provide experimental results to support the theory by showing that the spatio-temporal filtering induced by the focus of attention allows the system to globally transfer more information from the input stream over the focused areas and, in some contexts, over the whole frames with respect to the unfiltered case that yields uniform probability distributions.

We introduce manifold-learning flows (ℳ-flows), a new class of generative models that simultaneously learn the data manifold as well as a tractable probability density on that manifold. Combining aspects of normalizing flows, GANs, autoencoders, and energy-based models, they have the potential to represent data sets with a manifold structure more faithfully and provide handles on dimensionality reduction, denoising, and out-of-distribution detection. We argue why such models should not be trained by maximum likelihood alone and present a new training algorithm that separates manifold and density updates. In a range of experiments we demonstrate how ℳ-flows learn the data manifold and allow for better inference than standard flows in the ambient data space.

Sparse models are desirable for many applications across diverse domains as they can perform automatic variable selection, aid interpretability, and provide regularization. When fitting sparse models in a Bayesian framework, however, analytically obtaining a posterior distribution over the parameters of interest is intractable for all but the simplest cases. As a result practitioners must rely on either sampling algorithms such as Markov chain Monte Carlo or variational methods to obtain an approximate posterior. Mean field variational inference is a particularly simple and popular framework that is often amenable to analytically deriving closed-form parameter updates. When all distributions in the model are members of exponential families and are conditionally conjugate, optimization schemes can often be derived by hand. Yet, I show that using standard mean field variational inference can fail to produce sensible results for models with sparsity-inducing priors, such as the spike-and-slab. Fortunately, such pathological behavior can be remedied as I show that mixtures of exponential family distributions with non-overlapping support form an exponential family. In particular, any mixture of an exponential family of diffuse distributions and a point mass at zero to model sparsity forms an exponential family. Furthermore, specific choices of these distributions maintain conditional conjugacy. I use two applications to motivate these results: one from statistical genetics that has connections to generalized least squares with a spike-and-slab prior on the regression coefficients; and sparse probabilistic principal component analysis. The theoretical results presented here are broadly applicable beyond these two examples.

Quantization based on the binary codes is gaining attention because each quantized bit can be directly utilized for computations without dequantization using look-up tables. Previous attempts, however, only allow for integer numbers of quantization bits, which ends up restricting the search space for compression ratio and accuracy. In this paper, we propose an encryption algorithm/architecture to compress quantized weights so as to achieve fractional numbers of bits per weight. Decryption during inference is implemented by digital XOR-gate networks added into the neural network model while XOR gates are described by utilizing $\tanh(x)$ for backward propagation to enable gradient calculations. We perform experiments using MNIST, CIFAR-10, and ImageNet to show that inserting XOR gates learns quantization/encrypted bit decisions through training and obtains high accuracy even for fractional sub 1-bit weights. As a result, our proposed method yields smaller size and higher model accuracy compared to binary neural networks.

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$. We show first that the constraint matrix arising from the standard linear programming (LP) representation of the FS-WBP is \textit{not totally unimodular} when $m \geq 3$ and $n \geq 3$. This result resolves an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it proves that the FS-WBP in the standard LP form is not an MCF problem when $m \geq 3$ and $n \geq 3$. We also develop a provably fast \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of $\tilde{O}(mn^{7/3}\varepsilon^{-4/3})$, where $\varepsilon \in (0, 1)$ is the desired tolerance. This complexity bound is better than the best known complexity bound of $\tilde{O}(mn^2\varepsilon^{-2})$ for the IBP algorithm in terms of $\varepsilon$, and that of $\tilde{O}(mn^{5/2}\varepsilon^{-1})$ from accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of $n$. Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the \textsc{FastIBP} algorithm in practice.

Semi-supervised learning (SSL) provides an effective means of leveraging unlabeled data to improve a model’s performance. This domain has seen fast progress recently, at the cost of requiring more complex methods. In this paper we propose FixMatch, an algorithm that is a significant simplification of existing SSL methods. FixMatch first generates pseudo-labels using the model’s predictions on weakly-augmented unlabeled images. For a given image, the pseudo-label is only retained if the model produces a high-confidence prediction. The model is then trained to predict the pseudo-label when fed a strongly-augmented version of the same image. Despite its simplicity, we show that FixMatch achieves state-of-the-art performance across a variety of standard semi-supervised learning benchmarks, including 94.93% accuracy on CIFAR-10 with 250 labels and 88.61% accuracy with 40 – just 4 labels per class. We carry out an extensive ablation study to tease apart the experimental factors that are most important to FixMatch’s success. The code is available at https://github.com/google-research/fixmatch.

Market equilibrium is a solution concept with many applications such as digital ad markets, fair division, and resource sharing. For many classes of utility functions, equilibria can be captured by convex programs. We develop simple first-order methods suitable for solving these programs for large-scale markets. We focus on three practically-relevant utility classes: linear, quasilinear, and Leontief utilities. Using structural properties of market equilibria under each utility class, we show that the corresponding convex programs can be reformulated as optimization of a structured smooth convex function over a polyhedral set, for which projected gradient achieves linear convergence. To do so, we utilize recent linear convergence results under weakened strong-convexity conditions, and further refine the relevant constants in existing convergence results. Then, we show that proximal gradient (a generalization of projected gradient) with a practical linesearch scheme achieves linear convergence under the Proximal-PL condition, a recently developed error bound condition for convex composite problems. For quasilinear utilities, we show that Mirror Descent applied to a new convex program achieves sublinear last-iterate convergence and yields a form of Proportional Response dynamics, an elegant, interpretable algorithm for computing market equilibria originally developed for linear utilities. Numerical experiments show that Proportional Response is highly efficient for computing approximate market equilibria, while projected gradient with linesearch can be much faster when higher-accuracy solutions are needed.

In reinforcement learning, an agent attempts to learn high-performing behaviors through interacting with the environment, such behaviors are often quantified in the form of a reward function. However some aspects of behavior—such as ones which are deemed unsafe and to be avoided—are best captured through constraints. We propose a novel approach called First Order Constrained Optimization in Policy Space (FOCOPS) which maximizes an agent's overall reward while ensuring the agent satisfies a set of cost constraints. Using data generated from the current policy, FOCOPS first finds the optimal update policy by solving a constrained optimization problem in the nonparameterized policy space. FOCOPS then projects the update policy back into the parametric policy space. Our approach has an approximate upper bound for worst-case constraint violation throughout training and is first-order in nature therefore simple to implement. We provide empirical evidence that our simple approach achieves better performance on a set of constrained robotics locomotive tasks.

We propose firefly neural architecture descent, a general framework for progressively and dynamically growing neural networks to jointly optimize the networks' parameters and architectures. Our method works in a steepest descent fashion, which iteratively finds the best network within a functional neighborhood of the original network that includes a diverse set of candidate network structures. By using Taylor approximation, the optimal network structure in the neighborhood can be found with a greedy selection procedure. We show that firefly descent can flexibly grow networks both wider and deeper, and can be applied to learn accurate but resource-efficient neural architectures that avoid catastrophic forgetting in continual learning. Empirically, firefly descent achieves promising results on both neural architecture search and continual learning. In particular, on a challenging continual image classification task, it learns networks that are smaller in size but have higher average accuracy than those learned by the state-of-the-art methods.

We study an extension of the classic stochastic multi-armed bandit problem which involves multiple plays and Markovian rewards in the rested bandits setting. In order to tackle this problem we consider an adaptive allocation rule which at each stage combines the information from the sample means of all the arms, with the Kullback-Leibler upper confidence bound of a single arm which is selected in round-robin way. For rewards generated from a one-parameter exponential family of Markov chains, we provide a finite-time upper bound for the regret incurred from this adaptive allocation rule, which reveals the logarithmic dependence of the regret on the time horizon, and which is asymptotically optimal. For our analysis we devise several concentration results for Markov chains, including a maximal inequality for Markov chains, that may be of interest in their own right. As a byproduct of our analysis we also establish asymptotically optimal, finite-time guarantees for the case of multiple plays, and i.i.d. rewards drawn from a one-parameter exponential family of probability densities. Additionally, we provide simulation results that illustrate that calculating Kullback-Leibler upper confidence bounds in a round-robin way, is significantly more efficient than calculating them for every arm at each round, and that the expected regrets of those two approaches behave similarly.

Although Q-learning is one of the most successful algorithms for finding the best action-value function (and thus the optimal policy) in reinforcement learning, its implementation often suffers from large overestimation of Q-function values incurred by random sampling. The double Q-learning algorithm proposed in~\citet{hasselt2010double} overcomes such an overestimation issue by randomly switching the update between two Q-estimators, and has thus gained significant popularity in practice. However, the theoretical understanding of double Q-learning is rather limited. So far only the asymptotic convergence has been established, which does not characterize how fast the algorithm converges. In this paper, we provide the first non-asymptotic (i.e., finite-time) analysis for double Q-learning. We show that both synchronous and asynchronous double Q-learning are guaranteed to converge to an $\epsilon$-accurate neighborhood of the global optimum by taking $\tilde{\Omega}\left(\left( \frac{1}{(1-\gamma)^6\epsilon^2}\right)^{\frac{1}{\omega}} +\left(\frac{1}{1-\gamma}\right)^{\frac{1}{1-\omega}}\right)$ iterations, where $\omega\in(0,1)$ is the decay parameter of the learning rate, and $\gamma$ is the discount factor. Our analysis develops novel techniques to derive finite-time bounds on the difference between two inter-connected stochastic processes, which is new to the literature of stochastic approximation.

Stochastic Approximation (SA) is a popular approach for solving fixed-point equations where the information is corrupted by noise. In this paper, we consider an SA involving a contraction mapping with respect to an arbitrary norm, and show its finite-sample error bounds while using different stepsizes. The idea is to construct a smooth Lyapunov function using the generalized Moreau envelope, and show that the iterates of SA have negative drift with respect to that Lyapunov function. Our result is applicable in Reinforcement Learning (RL). In particular, we use it to establish the first-known convergence rate of the V-trace algorithm for off-policy TD-learning [18]. Importantly, our construction results in only a logarithmic dependence of the convergence bound on the size of the state-space.

We perform a careful, thorough, and large scale empirical study of the correspondence between wide neural networks and kernel methods. By doing so, we resolve a variety of open questions related to the study of infinitely wide neural networks. Our experimental results include: kernel methods outperform fully-connected finite-width networks, but underperform convolutional finite width networks; neural network Gaussian process (NNGP) kernels frequently outperform neural tangent (NT) kernels; centered and ensembled finite networks have reduced posterior variance and behave more similarly to infinite networks; weight decay and the use of a large learning rate break the correspondence between finite and infinite networks; the NTK parameterization outperforms the standard parameterization for finite width networks; diagonal regularization of kernels acts similarly to early stopping; floating point precision limits kernel performance beyond a critical dataset size; regularized ZCA whitening improves accuracy; finite network performance depends non-monotonically on width in ways not captured by double descent phenomena; equivariance of CNNs is only beneficial for narrow networks far from the kernel regime. Our experiments additionally motivate an improved layer-wise scaling for weight decay which improves generalization in finite-width networks. Finally, we develop improved best practices for using NNGP and NT kernels for prediction, including a novel ensembling technique. Using these best practices we achieve state-of-the-art results on CIFAR-10 classification for kernels corresponding to each architecture class we consider.

We consider a situation where an agent has $T$ ressources to be allocated to a larger number $N$ of actions. Each action can be completed at most once and results in a stochastic reward with unknown mean. The goal of the agent is to maximize her cumulative reward. Non trivial strategies are possible when side information on the actions is available, for example in the form of covariates. Focusing on a nonparametric setting, where the mean reward is an unknown function of a one-dimensional covariate, we propose an optimal strategy for this problem. Under natural assumptions on the reward function, we prove that the optimal regret scales as $O(T^{1/3})$ up to poly-logarithmic factors when the budget $T$ is proportional to the number of actions $N$. When $T$ becomes small compared to $N$, a smooth transition occurs. When the ratio $T/N$ decreases from a constant to $N^{-1/3}$, the regret increases progressively up to the $O(T^{1/2})$ rate encountered in continuum-armed bandits.